/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 In Exercises \(1-16,\) find a pa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 14

In Exercises \(1-16,\) find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) $$ \begin{array}{l}{\text { Tilted plane inside cylinder } \text { The portion of the plane }} \\ {x-y+2 z=2}\end{array} $$ $$ \begin{array}{l}{\text { a. Inside the cylinder } x^{2}+z^{2}=3} \\ {\text { b. Inside the cylinder } y^{2}+z^{2}=2}\end{array} $$

Short Answer

Expert verified
Parametrize: a. \((\sqrt{3}\cos(t), \sqrt{3}\cos(t)+2\sqrt{3}\sin(t)-2, \sqrt{3}\sin(t))\); b. \((\sqrt{2}\cos(t) - 2\sqrt{2}\sin(t) + 2, \sqrt{2}\cos(t), \sqrt{2}\sin(t))\).

Step by step solution

01

Understand the Problem

We need to find the parametrization of a tilted plane described by \(x-y+2z=2\) within a cylinder. We have two scenarios: a. the plane inside the cylinder \(x^2+z^2=3\) and b. the plane inside the cylinder \(y^2+z^2=2\).
02

Analyze the Plane and Cylinder a

First, look at the portion of the plane inside the cylinder \(x^2+z^2=3\). We need to express \(y\) in terms of \(x\) and \(z\) from \(x-y+2z=2\), which gives \(y = x + 2z - 2\). Given the circle of the cylinder in the x-z plane, \(x = \sqrt{3}\cos(t)\) and \(z = \sqrt{3}\sin(t)\), for \(t\) between \(0\) and \(2\pi\).
03

Parametrize the Surface for Cylinder a

Using \(x = \sqrt{3}\cos(t)\), \(z = \sqrt{3}\sin(t)\) and \(y = x + 2z - 2\), the parameterization becomes \((x(t), y(t), z(t)) = (\sqrt{3}\cos(t), \sqrt{3}\cos(t) + 2\sqrt{3}\sin(t) - 2, \sqrt{3}\sin(t))\).
04

Analyze the Plane and Cylinder b

Now analyze the plane inside the cylinder \(y^2+z^2=2\). We can still use \(y = x + 2z - 2\). The circle of the cylinder lies in the y-z plane, so \(y = \sqrt{2}\cos(t)\) and \(z = \sqrt{2}\sin(t)\). The \(x\) component is solved using \(x = y - 2z + 2\).
05

Parametrize the Surface for Cylinder b

Using \(y = \sqrt{2}\cos(t)\), \(z = \sqrt{2}\sin(t)\), and \(x = y - 2z + 2\), the parametrization becomes \((x(t), y(t), z(t)) = (\sqrt{2}\cos(t) - 2\sqrt{2}\sin(t) + 2, \sqrt{2}\cos(t), \sqrt{2}\sin(t))\), for \(t\) from \(0\) to \(2\pi\).
06

Verify the Parametrizations

We check that both parametrizations satisfy both the plane equation \(x-y+2z=2\) and are confined within their respective cylinders. For any value of \(t\), both parameterized representations should solve the initial conditions and define the surface correctly.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plane Equation
A plane equation is a mathematical expression that represents a flat, two-dimensional surface in three-dimensional space. Typically, it's written in the form \( Ax + By + Cz = D \), where \( A \), \( B \), and \( C \) are the coefficients of the plane, and \( D \) represents the plane's position relative to the origin.

In the original exercise, the plane equation is \( x-y+2z=2 \). This equation signifies a plane with specific characteristics:
  • The plane tilts through the coordinate axes, as denoted by the coefficients of \( x \), \( y \), and \( z \).
  • The constant 2 implies the plane doesn't pass through the origin but is shifted parallel from it.
This tilt shows how the plane inclines in the 3D space, giving a unique orientation that we need to confine inside different cylinders.
Cylinder Geometry
Cylinder geometry focuses on surfaces formed by revolving a line around an axis, maintaining a set radius. Two scenarios from the exercise showcase the cylindrical shapes confined in the x-z and y-z planes.

For the first cylinder \( x^2+z^2=3 \):
  • This shape is circular from the top view, centered at the origin with radius \( \sqrt{3} \).
  • Structurally, it extends indefinitely along the y-axis.

In the second cylinder \( y^2+z^2=2 \):
  • It has a circle viewed along the x-axis, with a radius of \( \sqrt{2} \).
  • The length of this cylinder extends along the x-axis instead.
These cylinders confine specific parts of the plane. Understanding their geometry is critical to finding the surface's correct parametrization.
Surface Parameterization
Surface parameterization involves using one or more parameters to describe a surface in mathematical terms. It's a powerful technique to transform the surface into a form that's easy to manipulate and understand.

For both scenarios in the exercise:
  • We set parameters \( t \) for the circular path within each cylinder.
  • Each point on the plane confined in the cylinder is mapped using functions of \( t \).
  • The parameterization helps us convert complex geometric descriptions into accessible coordinate functions.
With these tools, we ensure that each point satisfies both the plane and cylinder equations, enabling us to visualize and analyze the intersection between these surfaces intuitively.
Trigonometric Parametrization
Trigonometric parametrization uses trigonometric functions like sine and cosine because they naturally describe circular motion. In the given problem, equations with terms like \( \sqrt{3}\cos(t) \) and \( \sqrt{3}\sin(t) \) reflect these elements.

For scenario a, representing the circle within a cylinder in the x-z plane:
  • The parameters \( x = \sqrt{3}\cos(t) \) and \( z = \sqrt{3}\sin(t) \) capture the circle explicitly.

For scenario b, describing the cylinder in the y-z plane:
  • We use \( y = \sqrt{2}\cos(t) \) and \( z = \sqrt{2}\sin(t) \) for parametrization.
Through trigonometric functions, we efficiently trace the circles and satisfy geometric constraints within the cylinders over the interval \( t \) from \( 0 \) to \( 2\pi \).
Coordinate System
In three-dimensional space, coordinate systems define the placement of points using three numbers, typically \( (x, y, z) \). These systems are vital for expressing geometric figures and applying mathematical equations.

The problem emphasizes:
  • Coordinate transformation, where we go from 3D coordinates to a parameterized 1D representation.
  • Blending trigonometric representations with the plane equation to assure all points lie accurately within the desired shapes.
The coordinate system in this context is not just about locating the cylinder or plane but also aligning them in relation to the entire domain, ensuring clarity in visualizing each surface part.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Verify Stokes' Theorem for the vector field \(\mathbf{F}=2 x y \mathbf{i}+x \mathbf{j}+\) \((y+z) \mathbf{k}\) and surface \(z=4-x^{2}-y^{2}, z \geq 0,\) oriented with unit normal n pointing upward.

In Exercises 9-20, use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Cylinder and paraboloid \(\mathbf{F}=y \mathbf{i}+x y \mathbf{j}-z \mathbf{k}\) \(D :\) The region inside the solid cylinder \(x^{2}+y^{2} \leq 4\) between the plane \(z=0\) and the paraboloid \(z=x^{2}+y^{2}\)

The tangent plane at a point \(P_{0}\left(f\left(u_{0}, v_{0}\right), g\left(u_{0}, v_{0}\right), h\left(u_{0}, v_{0}\right)\right)\) on a parametrized surface \(\mathbf{r}(u, v)=f(u, v) \mathbf{i}+g(u, v) \mathbf{j}+h(u, v) \mathbf{k}\) is the plane through \(P_{0}\) normal to the vector \(\mathbf{r}_{u}\left(u_{0}, v_{0}\right) \times \mathbf{r}_{v}\left(u_{0}, v_{0}\right),\) the cross product of the tangent vectors \(\mathbf{r}_{u}\left(u_{0}, v_{0}\right)\) and \(\mathbf{r}_{v}\left(u_{0}, v_{0}\right)\) at \(P_{0} .\) In Exercises \(27-30,\) find an equation for the plane tangent to the surface at \(P_{0} .\) Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. $$ \begin{array}{l}{\text { Hemisphere } \quad \text { The hemisphere surface } \mathbf{r}(\phi, \theta)=(4 \sin \phi \cos \theta) \mathbf{i}} \\ {+(4 \sin \phi \sin \theta) \mathbf{j}+(4 \cos \phi) \mathbf{k}, 0 \leq \phi \leq \pi / 2,0 \leq \theta \leq 2 \pi} \\ {\text { at the point } P_{0}(\sqrt{2}, \sqrt{2}, 2 \sqrt{3}) \text { corresponding to }(\phi, \theta)=} \\ {(\pi / 6, \pi / 4)}\end{array} $$

In Exercises \(9-20,\) use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) $$\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$$ $$\begin{array}{l}{\text { a. Cube } D :\quad \quad \quad \quad \text { The cube cut from the first octant by }} \quad\quad\quad\quad\quad\quad\quad\quad {\text { the planes } x=1, y=1, \text { and } z=1}\end{array} $$ $$b. Cube D :\quad\quad\quad\quad \quad The \quad cube \quad bounded \quad \quad \quad by \quad the \quad\quad\quad\quad\quad\quad\quad\quad\quad planes=\pm 1, y=\pm 1, \text { and } z=\pm 1$$ $$\begin{array}{c}{\text { c. Cylindrical can } D : \text { The region cut from the solid cylinder }} \quad\quad\quad\quad\quad\quad\quad\quad {x^{2}+y^{2} \leq 4 \text { by the planes } z=0 \text { and }} \\\ {z=1}\end{array}$$

Gradient of a line integral Suppose that \(\mathbf{F}=\nabla f\) is a conservative vector field and $$g(x, y, z)=\int_{(0,0,0)}^{(x, y z)} \mathbf{F} \cdot d \mathbf{r}$$ Show that \(\nabla g=\mathbf{F}\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.